#354025

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(6,8)(13,14,18,21,17,20,26,28)(15,22,24,25,23,19,27,16);(2,4)(9,10,11,12)(13,15)(14,19)(16,21)(17,23)(18,24)(20,22)(25,28)(26,27);(5,6,7,8)(13,16,18,22,17,25,26,19)(14,15,21,24,20,23,28,27);(14,20)(15,23)(21,28)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261221);